Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

TL;DR

This paper characterizes the torsion subgroup of endo-trivial modules for finite groups with dihedral Sylow 2-subgroups, revealing its structure and exceptions, and provides a reduction method applicable at any prime.

Contribution

It offers a detailed description of the torsion subgroup of endo-trivial modules for groups with dihedral Sylow 2-subgroups, including new cases and a reduction technique for general primes.

Findings

For |P|≥8, TT(G) is isomorphic to the group of one-dimensional modules, except possibly for G/O_{2'}(G) ≅ A6.

For |P|=4, the structure of TT(G) is more complex but similarly characterized.

A reduction result relates TT(G) to TT(G/H) for normal p'-subgroups H, applicable at any prime.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and $G$ a finite group. We provide a description of the torsion subgroup $TT(G)$ of the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules when $p=2$ and $G$ has a dihedral Sylow $2$-subgroup $P$. We prove that, in the case $|P|\geq 8$, $TT(G)\cong X(G)$ the group of one-dimensional $kG$-modules, except possibly when $G/O_{2'}(G)\cong \mathfrak{A}_6$, the alternating group of degree $6$; in which case $G$ may have $9$-dimensional simple torsion endo-trivial modules. We also prove a similar result in the case $|P|=4$, although the situation is more involved. Our results complement the tame-representation type investigation of endo-trivial modules started by Carlson-Mazza-Th\'evenaz in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore we provide a general reduction result, valid at any prime $p$, to recover the structure of $TT(G)$ from the structure of $TT(G/H)$, where $H$ is a normal $p'$-subgroup of $G$.